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% Transformation and summation formulas for
% multiple hypergeometric series, and the 9-$j$ coefficient
% J. Van der Jeugt, S.N. Pitre, and K. Srinivasa Rao
% in Special Functions and Differential Equations
% Eds. K. Srinivasa Rao, R. Jagannathan, G. Vanden Berghe 
% and J. Van der Jeugt,
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\begin{document}
\begin{center}
{\Large \bf 
Transformation and summation formulas for\\[3mm]
multiple hypergeometric series, and the 9-$j$ coefficient
}\\[15mm]
{\bf J.\ Van der Jeugt\footnote{Research
Associate of the Fund for Scientific Research -- Flanders (Belgium).},
S.N.\ Pitre, and K.\ Srinivasa Rao\footnote{Permanent Address~:
Institute of Mathematical Sciences, CIT Campus, Madras 600 113, India.
}}\\[1cm] 
Department of Applied Mathematics and Computer Science,\\
University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium.\\[2mm]
E-mail : Joris.VanderJeugt@rug.ac.be; rao@imsc.ernet.in.
\end{center}

\vskip 15mm

\noindent
For the classical hypergeometric series ${}_pF_q$ of a single variable,
the most impressive results are the various summation theorems due to
Gauss, Vandermonde, Saalsch\"utz, Watson, Dougall, Dixon, and Bailey,
amongst many others. Just as important are the
transformation formulas expressing one hypergeometric series in terms
of a different one. For hypergeometric series
of two or more variables, few summation and transformation formulas as
known. The simplest double hypergeometric
series are the Appell series and the Horn series; more general double
series are the Kamp\'e de F\'eriet series and its generalisations,
which form the topic of this contribution.

These Kamp\'e de F\'eriet functions have appeared naturally in the
study of series expressions for the 9-$j$ coefficient of angular
momentum theory. In particular, by comparing the series expressions for
two symmetries of a certain 9-$j$ coefficient, summation and
transformation formulas for Kamp\'e de F\'eriet functions can be
obtained. This new approach has been very fruitful, and has given rise to a
large number of summation and transformation formulas for double
hypergeometric series of Kamp\'e de F\'eriet type, both terminating and
non-terminating, and with a large number of generic parameters.

In the present contribution, we summarize and illustrate some of these
new results. We also give the $q$-extension of certain transformation
and summation formulas.


\section{Multiple hypergeometric series}

In the theory of special functions the Gauss series, or ordinary
hypergeometric series~\cite{Bailey,Slater},
\beq
{}_2F_1\left( {a,b\atop c};x\right) = \sum_{k=0}^\infty
 {(a)_k(b)_k\over (c)_k} {x^k\over k!},
\label{gauss}
\eeq
plays a fundamental role. In (\ref{gauss}), we have used the common
notation for Pochhammer symbols~:
\beq
(a)_n = \left\{
 \begin{array}{ll}
 1, & \hbox{ if } n=0,\\
 a(a+1)\ldots (a+n-1), & \hbox{ if } n=1,2,3,\ldots;
 \end{array} \right.
\label{poch}
\eeq
or, in terms of Gamma functions,
\beq
(a)_n=\Gamma(a+n)/\Gamma(a), \qquad (-a\not\in \Zah_+).
\eeq
The series (\ref{gauss}) converges absolutely within the unit circle
$|x|<1$, and if $\Re(c-a-b)>0$ also for $|x|=1$. If one of the
numerator parameters $a$ or $b$ is a negative integer, the series is
terminating and the question of convergence does not enter.

The Gauss series has been extended to generalised hypergeometric
series~\cite{Bailey,Slater} 
depending upon several numerator parameters $a_i$ and denominator
parameters $b_i$, 
\beq
{}_pF_q\left( {a_1,\ldots,a_p\atop b_1,\ldots,b_q};x\right) = 
\sum_{k=0}^\infty {(a_1,\ldots,a_p)_k\over (b_1,\ldots,b_q)_k}
{x^k\over k!}, 
\label{hypser}
\eeq
where the common convention for Pochhammer symbols is used~:
\[
(a_1,\ldots,a_p)_k= (a_1)_k \cdots (a_p)_k.
\]
Here and hereafter, we shall assume that denominator parameters are
neither zero nor a negative integer. 

Hypergeometric series in several variables have also been studied in
literature~\cite{SriKar}. Standard series are the Appell series $F_1$,
$F_2$, $F_3$ and $F_4$~\cite{Appell}. For example,
\beq
F_3(a,a',b,b';c;x,y)= 
\sum_{k,l=0}^\infty {(a,b)_k(a',b')_l\over (c)_{k+l}}
{x^k\over k!} {y^l\over l!},
\eeq
where $\max(|x|,|y|)\leq 1$ for convergence. Following Appell,
Horn~\cite{Horn} 
defined a number of different hypergeometric series in two variables.
The Appell series were generalised by Kamp\'e de F\'eriet~\cite{K1,K2},
who defined 
a very general hypergeometric series in two variables. We give here a
slightly more general series, in the literature still referred to as
the Kamp\'e de F\'eriet series~\cite{SriKar}~:
\bea
&&F^{A:B;B'}_{C:D;D'}\left( {a_1,\ldots,a_A \atop c_1,\ldots, c_C} :
{b_1,\ldots,b_B \atop d_1,\ldots, d_D}; 
{b_1',\ldots,b_{B'}' \atop d_1',\ldots, d_{D'}'}; x,y \right) = \nn\\
&&\qquad \sum_{k,l=0}^\infty {(a_1,\ldots,a_A)_{k+l} \over (c_1,\ldots,
c_C)_{k+l}} {(b_1,\ldots,b_B)_k \over (d_1,\ldots, d_D)_l} 
{(b_1',\ldots,b_{B'}')_l \over (d_1',\ldots, d_{D'}')_l} {x^k\over k!}
{y^l\over l!}.
\label{kdf}
\eea
Other generalisations to more variables have been given by
Lauricalla~\cite{Lauri}, Srivastava and Daoust~\cite{SriDao}.

Among the famous results in hypergeometric function theory are the
summation and transformation formulas. Gauss's summation formula
reads~\cite{Bailey,Slater} 
\beq
{}_2F_1\left( {a,b\atop c};1\right) =
{\Gamma(c,c-a-b)\over\Gamma(c-a,c-b)} \qquad \Re(c-a-b)>0.
\eeq
Herein, we have used the convention
\[
\Gamma(a,b,c,\ldots)=\Gamma(a)\Gamma(b)\Gamma(c)\cdots.
\]
For a ${}_3F_2(1)$ of unit argument, we have Saalsch\"utz's summation
formula~\cite{Slater}~: 
\beq
{}_3F_2\left( {a,b,-n\atop c,d};1\right) =
{(c-a,c-b)_n\over(c,c-a-b)_n},
\eeq
where $c+d=a+b-n+1$ (and $n\in\Zah_+$). 
There are many other well known summation
theorems; for a more extended list, see e.g.~\cite{Slater}. 

Transformation formulas express one hypergeometric series in terms of
another one. Dixon has given a general transformation formula for a
${}_3F_2(1)$, and Thomae for a terminating ${}_3F_2(1)$~:
\beq
{}_3F_2\left( {a,b,-n\atop d,e};1\right) = {(d-a)_n\over (d)_n}
{\ }_3F_2\left( {a,e-b,-n\atop 1+a-d-n,e};1\right).
\eeq
Another often used transformation formula is due to Bailey and applies
to a terminating Saalsch\"utzian ${}_4F_3(1)$ series~\cite{Bailey}.

When it comes to summation or transformation formulas of multiple
hypergeometric series, few results are known. For series of Kamp\'e de
F\'eriet type, we have found two summation formulas in the literature.
One is due to Jain~\cite{Jain} (see also~Srivastava~\cite{Sri})~:
\beq
F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,-n\atop
1+a+b-n-d};{d-a,d-b,-m\atop 1-a-b-m+d};1,1\right)= 
{(d-a,d-b)_{n}(a,b)_{m}\over(d)_{n+m}(d-a-b)_{n}
(a+b-d)_{m} }.
\label{jain}
\eeq
The second one is known as Carlitz's identity~\cite{Carl}~:
\beq
F_{0:2;2}^{1:2;2}\left( {e\atop\hy }:{a,-n\atop
c,e};{1+a-c-n,-m\atop 2-c-n-m,e};1,1\right)= 
{(e)_{m+n}(c-a)_{n}(a)_{m}\over(c,e)_{n}
(c+n-1,e)_{m} }.
\label{carlitz}
\eeq
Note that both summation formulas are terminating ($n,m\in\Zah_+$), and
have only three generic parameters.

\section{The 9-$j$ coefficient and new summation and transformation
formulas} 

In the quantum theory of angular momentum~\cite{BieLou,RaoRaj}, or
representation theory of 
$su(2)$, the Clebsch-Gordan coefficients play an essential role. These
Clebsch-Gordan coefficients, proportional to 3-$j$ coefficients, can be
expressed in terms of series of type ${}_3F_2(1)$. The Racah
coefficients, or 6-$j$ coefficients, are transformation coefficients
relating two orthogonal bases corresponding to two ``coupling schemes''
in the tensor product of three irreducible representations. These
coefficients can be expressed as a single sum expression, corresponding
to a Saalsch\"utzian ${}_4F_3(1)$ series. When considering the tensor
product of four representations, and two coupling schemes, the
corresponding basis vectors are related by means of a 9-$j$
coefficient, denoted by
\beq
\left\{ 
\begin{array}{ccc}
 j_1 & j_2 & j_{12} \\
 j_3 & j_4 & j_{34} \\
j_{13} & j_{24} & j \end{array} \right\}.
\eeq
Several expressions exist for a 9-$j$ coefficient. The one that is
most commonly used expresses the 9-$j$ coefficient as a single sum over a
product of three 6-$j$ coefficients; thus in total this is a fourfold
sum. A less known expression for the 9-$j$ coefficient is due to
Ali\v sauskas, Jucys and Bandzaitis~\cite{Ali,JucBan}. Their formula is
a (rather 
complicated) triple sum series of hypergeometric type for the 9-$j$
coefficient. A typical 
feature of this triple sum series is that it does not yield the 72
symmetries of the 9-$j$ coefficient. This inherent lack of symmetry has
been turned to advantage, being the source for obtaining new relations
between hypergeometric series. Because the general triple sum series is
complicated, it is advantageous to work with series for so-called
stretched 9-$j$ coefficients; these have less parameters and thus also
reduce and simplify the general triple sum series. 

Let us briefly describe one example. From classical symmetries, it
follows that the following 9-$j$ coefficients are equal~:
\[
\left\{ \begin{array}{ccc}
 j & j_{24} & j_{1}+j_3 \\
 j_{12} & j_2 & j_{1} \\
j_{34} & j_3+j_{34} & j_3 \end{array} \right\}=
\left\{ \begin{array}{ccc}
 j & j_{34} & j_{12} \\
 j_1+j_3 & j_3 & j_1 \\
j_{24} & j_3+j_{34} & j_2 \end{array} \right\}.
\]
Using the triple sum series, the lhs of this equation reduces to a
double sum, and the rhs reduces to a single sum. In fact, the double
sum can be written in terms of a Kamp\'e de F\'eriet series of type
$F^{0:3;3}_{1:1;1}(1,1)$, and the single sum in terms of an ordinary
${}_4F_3(1)$ series. Thus the above equality implies a transformation
formula between these two series. 

Using this idea, several new transformation and summation formulas for
double hypergeometric series of Kamp\'e de F\'eriet type have been
obtained~\cite{jpa1,jpa2,kdf1,kdf2}. In the following section we shall
give some examples. 

\section{Examples and $q$-extensions}

One of the transformation formulas that has been obtained this way is
the following
\bea
&&F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,c\atop
e};{a',b',c'\atop e'};1,1\right) = 
{ \Gamma(d,e',d+e'-a'-b'-c') \over
\Gamma(a',d+e'-a'-b',d+e'-a'-c')}\nn\\[1mm] 
&&\qquad\qquad\qquad\times {\ }_4F_3\left({a,e-b,e-c,d+e'
-a'-b'-c' \atop e,d+e'-a'-b', d+e'-a'-c'};1\right),
\label{kdf1} \\[1mm]
&&\qquad\hbox{ when } a'=d-a \hbox{ and } e'=d+e-a-b-c. \nn
\eea
For convergence, we must have $\Re(a')>0$, $\Re(e')>0$ and
$\Re(d+e'-a'-b'-c')>0$. A general proof of (\ref{kdf1}) has been given
in~\cite{kdf1}. Once such a transformation formula has been derived, a
number of summation formulas can also be obtained by further
specialising the parameters. Two examples that follow immediately from
(\ref{kdf1}) are
\bea
&&F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,c\atop
e};{d-a,d-b,-c\atop d+e-a-b-c};1,1\right) = 
{ \Gamma(e,d+e-a-b-c) \over
\Gamma(e-c,d+e-a-b)},
\label{sm1} \\[1mm]
&&\qquad\hbox{ for } \Re(e)>0 \hbox{ and } \Re(d+e-a-b-c)>0; \nn
\eea
and
\bea
&&F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,c\atop
e};{d-a,d-b,d-c\atop d+e-a-b-c};1,1\right) = 
{ \Gamma(e,e-d,d+e-a-b-c) \over
\Gamma(e-a,e-b,e-c)},
\label{sm2} \\[1mm]
&&\qquad\hbox{ for } \Re(e-d)>0 \hbox{ and } \Re(d+e-a-b-c)>0, \nn
\eea
because under these specialisations the ${}_4F_3(1)$ becomes a
${}_2F_1(1)$ which can be summed using Gauss's summation theorem.
Note that (\ref{sm1}) and (\ref{sm2}) are for non-terminating series,
with five generic parameters. 

The proof of (\ref{kdf1}) -- which was inspired by Gasper's
proof~\cite{Gasper} of one of our earlier results -- uses a number of
classical results, such as 
the beta function integral representation and Euler's identity. All
these classical results have a $q$-extension, so one can go through the
proof of (\ref{kdf1}) step by step, and try to find its
$q$-extension. Let $0<q<1$, 
and define the $q$-shifted factorial as follows~\cite{GasRah}~:
\beq
(a;q)_n = \left\{
 \begin{array}{ll}
 1, & \hbox{ if } n=0,\\
 (1-a)(1-aq)\cdots (1-aq^{n-1}), & \hbox{ if } n=1,2,3,\ldots;
 \end{array} \right.
\label{qfact}
\eeq
similarly, one has
\beq
(a;q)_\infty = \prod_{k=0}^\infty (1-a q^k).
\eeq
The $q$-extension of ${}_{r+1}F_r$ is defined as follows~\cite{GasRah}:
\beq
{}_{r+1}\vp_r\left( {a_1,\ldots,a_{r+1}\atop b_1,\ldots,b_r};q;x\right) = 
\sum_{k=0}^\infty {(a_1,\ldots,a_{r+1};q)_k\over (q,b_1,\ldots,b_r;q)_k}
x^k, 
\label{qhypser}
\eeq
where
\[
(a_1,\ldots,a_{r+1};q)_k = (a_1;q)_k \cdots (a_{r+1};q)_k.
\]

In order to have analogous results for double $q$-series, we were
forced to introduce the following definition~:
\beq
\vp^{0:3;3}_{1:1;1} \left( {\hy\atop d}: {a,b,c\atop e}; {a',b',c'
\atop e'}; q,x,y \right) = 
\sum_{k,l=0}^\infty {(a,b,c;q)_k (a',b',c';q)_l \over (d;q)_{k+l}
(q,e;q)_k (q,e';q)_l} q^{kl} x^k y^l.
\eeq
Note that the factor $q^{kl}$ is somewhat surprising. With this
definition, the $q$-extension of (\ref{kdf1}) can be obtained~: for
$a'=d/a$ and $e'=de/abc$, we have
\bea
&&\vp^{0:3;3}_{1:1;1} \left( {\hy\atop d}: {a,b,c\atop e}; {a',b',c'
\atop e'}; q,{de\over abc},{de'\over a'b'c'} \right) = 
 {(a',de'/a'b', de'/a'c';q)_\infty \over
(d,de/abc,de'/a'b'c';q)_\infty} \nn\\[1mm]
&&\qquad\qquad\qquad\times {\ }_4\vp_3\left( {a,e/b,e/c,de'/a'b'c'
\atop e,de'/a'b', de'/a'c'} ; q, a' \right).
\eea
Summation formulas then follow by making similar specialisations as in
(\ref{kdf1}). For example, the $q$-extension of (\ref{sm2}) becomes,
\beq
\vp^{0:3;3}_{1:1;1} \left( {\hy\atop d}: {a,b,c\atop e}; {d/a,d/b,d/c
\atop de/abc}; q,{de\over abc},{e\over d} \right) =
{(e/a,e/b,e/c;q)_\infty \over (e,e/d,de/abc;q)_\infty}.
\eeq

Here, we have given only one transformation formula and two summation
formulas for Kamp\'e de F\'eriet series $F^{0:3;3}_{1:1;1}(1,1)$ and
their $q$-extensions. From a study of more 9-$j$ coefficient
series expressions, many other transformation and summation formulas
have been obtained. In particular, for series of type
$F^{0:3;3}_{1:1;1}(1,1)$, we obtained three distinct transformation
formulas, and a list of 7 summation formulas (one of which is the Jain
formula (\ref{jain}))~\cite{kdf1}. Three transformation formulas were
also deduced for series of type $F^{0:3;4}_{1:1;2}(1,1)$. 
For series of type
$F^{1:2;2}_{0:2;2}(1,1)$, we obtained four distinct transformation
formulas, and again a list of 7 summation formulas (one of which is
Carlitz's identity (\ref{carlitz}))~\cite{kdf2}.

\section*{Acknowledgements}

This research was partly supported by the E.E.C. (contract No.
CI1*-CT92-0101).

%\newpage

\begin{thebibliography}{99}
\bibitem{Ali}
{S.J.\ Ali\v sauskas and A.P.\ Jucys},
%Weight lowering operators and the mul\-ti\-pli\-ci\-ty-free isoscalar
%factors for the group $R_5$,
{\em J. Math. Phys.} {\bf 12} (1971) 594--605.

\bibitem{Appell}
P.\ Appell,
%Sur les s\'eries hyperg\'eom\'etriques de deux variables, et sur des
%\'equations diff\'erentielles lin\'eaires aux d\'eriv\'ees partielles,
{\em C.R.\ Acad.\ Sci.\ Paris} {\bf 90} (1880) 296--298.

\bibitem{K2}
{ P.\ Appell and J.\ Kamp\'e de F\'eriet},
{\em Fonctions hyperg\'eom\'etriques et hy\-per\-sph\'e\-ri\-ques} (Gauthier
Villars, Paris, 1926).

\bibitem{Bailey}
{W.N.\ Bailey}, {\em Generalized Hypergeometric Series}
(Cambridge Univ.\ Press, Cambridge, 1935).

\bibitem{BieLou}
{L.C.\ Biedenharn and J.D.\ Louck}, {\em Angular Momentum in Quantum
Physics} and {\em The Racah-Wigner algebra in Quantum Theory}
(Addison-Wesley, Reading, 1981). 

\bibitem{Carl}
L.\ Carlitz, 
%Summation of a double hypergeometric series, 
{\em Matematiche (Catania)} {\bf 22} (1967) 138--142.

\bibitem{Gasper}
{G.\ Gasper}, private communication.

\bibitem{GasRah}
{G.\ Gasper and M.\ Rahman}, {\em Basic hypergeometric series}
(Cambridge Univ.\ Press, Cambridge, 1990).

\bibitem{Horn}
J.\ Horn,
%Hypergeometrische Funktionen zweier Ver\"anderlichen,
{\em Math.\ Ann.} {\bf 105} (1931) 381--407.

\bibitem{Jain}
R.N.\ Jain, 
%Sum of a double hypergeometric series,  
{\em Matematiche (Catania)} {\bf 21} (1966) 300--301.

\bibitem{JucBan}
{A.P.\ Jucys and A.A.\ Bandzaitis}, {\em Angular Momentum in Quantum
Physics} (Mokslas, Vilnius, 1977).

\bibitem{K1}
J.\ Kamp\'e de F\'eriet, 
%Les fonctions hyperg\'eom\'etriques d'ordre sup\'erieur \'a deux
%variables, 
{\em C.\ R.\ Acad.\ Sci.\ Paris} {\bf 173} (1921) 401--404.

\bibitem{Lauri}
G.\ Lauricella,
%Sulle funzioni ipergeometriche a pi\`u variabili,
{\em Rend.\ Circ.\ Mat.\ Palermo} {\bf 7} (1893) 111--158.

\bibitem{kdf1}
S.N.\ Pitre and J.\ Van der Jeugt, 
%Transformation and summation formulas for Kamp\'e de F\'eriet series
%$F^{0:3}_{1:1}(1,1)$, 
{\em J.\ Math.\ Anal.\ Appl.} {\bf 202} (1996), 121--132.

\bibitem{Slater}
{L.J.\ Slater}, {\em Generalized hypergeometric functions}
(Cambridge Univ.\ Press, Cambridge, 1966).

\bibitem{jpa1}
K.\ Srinivasa Rao and J.\ Van der Jeugt, 
%Stretched 9-$j$ coefficients and summation theorems, 
{\em J.\ Phys.\ A} {\bf 27} (1994) 3083--3090. 

\bibitem{RaoRaj}
{K.\ Srinivasa Rao and V.\ Rajeswari}, {\em Quantum Theory of Angular
Momentum : Selected Topics} (Narosa Publishing House, New
Delhi, 1993).

\bibitem{Sri}
H.M.\ Srivastava, 
%Sums of certain double $q$-hypergeometric series, 
{\em Rend.\ Sem.\ Mat.\ Univ.\ Padova} {\bf 72} (1984) 1--8.

\bibitem{SriDao}
H.M.\ Srivastava and M.C.\ Daoust,
%Certain generalized Neumann expansions associated with the Kamp\'e de
%F\'eriet function,
{\em Indag.\ Math.} {\bf 31} (1969) 449--457.

\bibitem{SriKar}
{H.M.\ Srivastava and P.W.\ Karlsson}, {\em Multiple Gaussian
Hypergeometric Series} (Halsted, New York, 1985).

\bibitem{jpa2}
{J.\ Van der Jeugt, S.N.\ Pitre and K.\ Srinivasa Rao}, 
%Multiple hypergeometric functions and 9-$j$ coefficients, 
{\em J. Phys. A~: Math. Gen.} {\bf 27} (1994) 5251--5264.

\bibitem{kdf2}
{J.\ Van der Jeugt, S.N.\ Pitre and K.\ Srinivasa Rao}, 
Transformation and summation formulas for double hypergeometric series,
University of Ghent preprint (1996).

\end{thebibliography}
\end{document}
